3.9.85 \(\int \frac {4 b+a x^5}{(-b+a x^5) \sqrt [4]{-b+c x^4+a x^5}} \, dx\)

Optimal. Leaf size=67 \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a x^5-b+c x^4}}\right )}{\sqrt [4]{c}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a x^5-b+c x^4}}\right )}{\sqrt [4]{c}} \]

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Rubi [F]  time = 1.50, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {4 b+a x^5}{\left (-b+a x^5\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4*b + a*x^5)/((-b + a*x^5)*(-b + c*x^4 + a*x^5)^(1/4)),x]

[Out]

Defer[Int][(-b + c*x^4 + a*x^5)^(-1/4), x] - b^(1/5)*Defer[Int][1/((b^(1/5) - a^(1/5)*x)*(-b + c*x^4 + a*x^5)^
(1/4)), x] - b^(1/5)*Defer[Int][1/((b^(1/5) + (-1)^(1/5)*a^(1/5)*x)*(-b + c*x^4 + a*x^5)^(1/4)), x] - b^(1/5)*
Defer[Int][1/((b^(1/5) - (-1)^(2/5)*a^(1/5)*x)*(-b + c*x^4 + a*x^5)^(1/4)), x] - b^(1/5)*Defer[Int][1/((b^(1/5
) + (-1)^(3/5)*a^(1/5)*x)*(-b + c*x^4 + a*x^5)^(1/4)), x] - b^(1/5)*Defer[Int][1/((b^(1/5) - (-1)^(4/5)*a^(1/5
)*x)*(-b + c*x^4 + a*x^5)^(1/4)), x]

Rubi steps

\begin {align*} \int \frac {4 b+a x^5}{\left (-b+a x^5\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx &=\int \left (\frac {1}{\sqrt [4]{-b+c x^4+a x^5}}+\frac {5 b}{\left (-b+a x^5\right ) \sqrt [4]{-b+c x^4+a x^5}}\right ) \, dx\\ &=(5 b) \int \frac {1}{\left (-b+a x^5\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx+\int \frac {1}{\sqrt [4]{-b+c x^4+a x^5}} \, dx\\ &=(5 b) \int \left (-\frac {1}{5 b^{4/5} \left (\sqrt [5]{b}-\sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}}-\frac {1}{5 b^{4/5} \left (\sqrt [5]{b}+\sqrt [5]{-1} \sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}}-\frac {1}{5 b^{4/5} \left (\sqrt [5]{b}-(-1)^{2/5} \sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}}-\frac {1}{5 b^{4/5} \left (\sqrt [5]{b}+(-1)^{3/5} \sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}}-\frac {1}{5 b^{4/5} \left (\sqrt [5]{b}-(-1)^{4/5} \sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}}\right ) \, dx+\int \frac {1}{\sqrt [4]{-b+c x^4+a x^5}} \, dx\\ &=-\left (\sqrt [5]{b} \int \frac {1}{\left (\sqrt [5]{b}-\sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx\right )-\sqrt [5]{b} \int \frac {1}{\left (\sqrt [5]{b}+\sqrt [5]{-1} \sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx-\sqrt [5]{b} \int \frac {1}{\left (\sqrt [5]{b}-(-1)^{2/5} \sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx-\sqrt [5]{b} \int \frac {1}{\left (\sqrt [5]{b}+(-1)^{3/5} \sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx-\sqrt [5]{b} \int \frac {1}{\left (\sqrt [5]{b}-(-1)^{4/5} \sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx+\int \frac {1}{\sqrt [4]{-b+c x^4+a x^5}} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.54, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4 b+a x^5}{\left (-b+a x^5\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(4*b + a*x^5)/((-b + a*x^5)*(-b + c*x^4 + a*x^5)^(1/4)),x]

[Out]

Integrate[(4*b + a*x^5)/((-b + a*x^5)*(-b + c*x^4 + a*x^5)^(1/4)), x]

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IntegrateAlgebraic [A]  time = 3.12, size = 67, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+c x^4+a x^5}}\right )}{\sqrt [4]{c}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+c x^4+a x^5}}\right )}{\sqrt [4]{c}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(4*b + a*x^5)/((-b + a*x^5)*(-b + c*x^4 + a*x^5)^(1/4)),x]

[Out]

(-2*ArcTan[(c^(1/4)*x)/(-b + c*x^4 + a*x^5)^(1/4)])/c^(1/4) - (2*ArcTanh[(c^(1/4)*x)/(-b + c*x^4 + a*x^5)^(1/4
)])/c^(1/4)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^5+4*b)/(a*x^5-b)/(a*x^5+c*x^4-b)^(1/4),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{5} + 4 \, b}{{\left (a x^{5} + c x^{4} - b\right )}^{\frac {1}{4}} {\left (a x^{5} - b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^5+4*b)/(a*x^5-b)/(a*x^5+c*x^4-b)^(1/4),x, algorithm="giac")

[Out]

integrate((a*x^5 + 4*b)/((a*x^5 + c*x^4 - b)^(1/4)*(a*x^5 - b)), x)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{5}+4 b}{\left (a \,x^{5}-b \right ) \left (a \,x^{5}+c \,x^{4}-b \right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^5+4*b)/(a*x^5-b)/(a*x^5+c*x^4-b)^(1/4),x)

[Out]

int((a*x^5+4*b)/(a*x^5-b)/(a*x^5+c*x^4-b)^(1/4),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{5} + 4 \, b}{{\left (a x^{5} + c x^{4} - b\right )}^{\frac {1}{4}} {\left (a x^{5} - b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^5+4*b)/(a*x^5-b)/(a*x^5+c*x^4-b)^(1/4),x, algorithm="maxima")

[Out]

integrate((a*x^5 + 4*b)/((a*x^5 + c*x^4 - b)^(1/4)*(a*x^5 - b)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {a\,x^5+4\,b}{\left (b-a\,x^5\right )\,{\left (a\,x^5+c\,x^4-b\right )}^{1/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*b + a*x^5)/((b - a*x^5)*(a*x^5 - b + c*x^4)^(1/4)),x)

[Out]

int(-(4*b + a*x^5)/((b - a*x^5)*(a*x^5 - b + c*x^4)^(1/4)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{5} + 4 b}{\left (a x^{5} - b\right ) \sqrt [4]{a x^{5} - b + c x^{4}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**5+4*b)/(a*x**5-b)/(a*x**5+c*x**4-b)**(1/4),x)

[Out]

Integral((a*x**5 + 4*b)/((a*x**5 - b)*(a*x**5 - b + c*x**4)**(1/4)), x)

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